L. Berman and J. Hartmanis conjectured that there is a polynomial-time computable isomorphism between any two languages m-complete ('Karp' complete) for NP. D. Joseph and P. Young discovered a structurally defined class of NP-complete sets and conjectured that certain of these sets are not isomorphic to the standard NP-complete sets for some one-way functions f. These two conjectures cannot both be correct. We introduce a new family of strong one-way functions, the scrambling functions. If f is a scrambling function, then the above mentioned sets are not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis conjecture fails. As evidence for the existence of scrambling functions, we show that much more powerful one-way functions, the annihilating functions, exist relative to a random oracle.